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Properties

Label 6027.2.a.bk
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 19 x^{12} + 36 x^{11} + 134 x^{10} - 237 x^{9} - 438 x^{8} + 716 x^{7} + 662 x^{6} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{6} - \beta_{5} - \beta_{2} + \cdots - 1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{6} - \beta_{5} - \beta_{2} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{12} - \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 14 q^{3} + 14 q^{4} - 10 q^{5} - 2 q^{6} - 6 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 14 q^{3} + 14 q^{4} - 10 q^{5} - 2 q^{6} - 6 q^{8} + 14 q^{9} - 3 q^{10} - 16 q^{11} + 14 q^{12} - 21 q^{13} - 10 q^{15} + 22 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{19} - 40 q^{20} + q^{22} - 7 q^{23} - 6 q^{24} + 22 q^{25} - 2 q^{26} + 14 q^{27} - 16 q^{29} - 3 q^{30} - 8 q^{31} - 19 q^{32} - 16 q^{33} - 33 q^{34} + 14 q^{36} + q^{37} - 32 q^{38} - 21 q^{39} + 13 q^{40} - 14 q^{41} + 14 q^{43} - 36 q^{44} - 10 q^{45} - 12 q^{46} - 12 q^{47} + 22 q^{48} - q^{50} - 12 q^{51} - 60 q^{52} - 20 q^{53} - 2 q^{54} + 11 q^{55} - 2 q^{57} + 21 q^{58} - 25 q^{59} - 40 q^{60} - 26 q^{61} + 33 q^{62} + 42 q^{64} - 8 q^{65} + q^{66} - 22 q^{67} - 15 q^{68} - 7 q^{69} - 36 q^{71} - 6 q^{72} - 31 q^{73} - 65 q^{74} + 22 q^{75} + 2 q^{76} - 2 q^{78} + 12 q^{79} - 112 q^{80} + 14 q^{81} + 2 q^{82} - 20 q^{83} + 40 q^{85} - 9 q^{86} - 16 q^{87} - 54 q^{88} - 39 q^{89} - 3 q^{90} + 63 q^{92} - 8 q^{93} - 14 q^{94} - 55 q^{95} - 19 q^{96} - 18 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 19 x^{12} + 36 x^{11} + 134 x^{10} - 237 x^{9} - 438 x^{8} + 716 x^{7} + 662 x^{6} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 115 \nu^{13} + 8468 \nu^{12} - 24595 \nu^{11} - 121798 \nu^{10} + 409674 \nu^{9} + 523923 \nu^{8} + \cdots - 272668 ) / 18364 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 66 \nu^{13} - 2845 \nu^{12} + 8640 \nu^{11} + 47189 \nu^{10} - 131524 \nu^{9} - 270514 \nu^{8} + \cdots + 145640 ) / 9182 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 433 \nu^{13} - 186 \nu^{12} - 11191 \nu^{11} + 11712 \nu^{10} + 97154 \nu^{9} - 152093 \nu^{8} + \cdots - 25044 ) / 18364 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 433 \nu^{13} + 186 \nu^{12} + 11191 \nu^{11} - 11712 \nu^{10} - 97154 \nu^{9} + 152093 \nu^{8} + \cdots + 61772 ) / 18364 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 657 \nu^{13} + 8616 \nu^{12} - 9569 \nu^{11} - 132302 \nu^{10} + 265034 \nu^{9} + 661925 \nu^{8} + \cdots - 166252 ) / 18364 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1101 \nu^{13} + 4502 \nu^{12} + 16835 \nu^{11} - 80292 \nu^{10} - 80530 \nu^{9} + 514897 \nu^{8} + \cdots - 12872 ) / 18364 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2301 \nu^{13} - 7032 \nu^{12} - 42039 \nu^{11} + 138006 \nu^{10} + 266526 \nu^{9} - 1000485 \nu^{8} + \cdots - 9076 ) / 18364 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 720 \nu^{13} - 1518 \nu^{12} - 13286 \nu^{11} + 28201 \nu^{10} + 85888 \nu^{9} - 191269 \nu^{8} + \cdots + 13459 ) / 4591 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2470 \nu^{13} + 4825 \nu^{12} + 46216 \nu^{11} - 85969 \nu^{10} - 315048 \nu^{9} + 554328 \nu^{8} + \cdots + 57572 ) / 9182 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5521 \nu^{13} - 14586 \nu^{12} - 87939 \nu^{11} + 233648 \nu^{10} + 475158 \nu^{9} - 1272953 \nu^{8} + \cdots + 41296 ) / 18364 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12521 \nu^{13} - 27814 \nu^{12} - 219659 \nu^{11} + 464720 \nu^{10} + 1391798 \nu^{9} + \cdots - 261192 ) / 18364 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - \beta_{12} + 2\beta_{11} + \beta_{10} + 10\beta_{6} + 10\beta_{5} + 10\beta_{2} + 31\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{13} + 13 \beta_{12} + 12 \beta_{11} + \beta_{10} - 11 \beta_{9} - 12 \beta_{8} - 3 \beta_{7} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{13} - 9 \beta_{12} + 26 \beta_{11} + 13 \beta_{10} + \beta_{9} + \beta_{8} - 3 \beta_{7} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16 \beta_{13} + 127 \beta_{12} + 112 \beta_{11} + 18 \beta_{10} - 94 \beta_{9} - 113 \beta_{8} + \cdots + 530 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 90 \beta_{13} - 53 \beta_{12} + 254 \beta_{11} + 126 \beta_{10} + 15 \beta_{9} + 8 \beta_{8} + \cdots + 671 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 176 \beta_{13} + 1114 \beta_{12} + 957 \beta_{11} + 209 \beta_{10} - 738 \beta_{9} - 968 \beta_{8} + \cdots + 3419 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 657 \beta_{13} - 200 \beta_{12} + 2234 \beta_{11} + 1103 \beta_{10} + 151 \beta_{9} + 6 \beta_{8} + \cdots + 4817 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1663 \beta_{13} + 9261 \beta_{12} + 7839 \beta_{11} + 2040 \beta_{10} - 5589 \beta_{9} - 7893 \beta_{8} + \cdots + 22732 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4513 \beta_{13} + 238 \beta_{12} + 18655 \beta_{11} + 9207 \beta_{10} + 1282 \beta_{9} - 650 \beta_{8} + \cdots + 34017 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.73044
2.55282
2.13215
1.64093
1.21066
0.766034
0.442251
0.181424
−0.712717
−0.766355
−1.67187
−1.68292
−2.13271
−2.69016
−2.73044 1.00000 5.45531 −3.67975 −2.73044 0 −9.43451 1.00000 10.0473
1.2 −2.55282 1.00000 4.51690 −1.95019 −2.55282 0 −6.42520 1.00000 4.97848
1.3 −2.13215 1.00000 2.54608 2.91322 −2.13215 0 −1.16433 1.00000 −6.21144
1.4 −1.64093 1.00000 0.692668 0.0817657 −1.64093 0 2.14525 1.00000 −0.134172
1.5 −1.21066 1.00000 −0.534291 −0.826979 −1.21066 0 3.06818 1.00000 1.00119
1.6 −0.766034 1.00000 −1.41319 0.561773 −0.766034 0 2.61462 1.00000 −0.430337
1.7 −0.442251 1.00000 −1.80441 −4.29274 −0.442251 0 1.68250 1.00000 1.89847
1.8 −0.181424 1.00000 −1.96709 3.75467 −0.181424 0 0.719724 1.00000 −0.681187
1.9 0.712717 1.00000 −1.49203 −0.415066 0.712717 0 −2.48883 1.00000 −0.295824
1.10 0.766355 1.00000 −1.41270 −1.12787 0.766355 0 −2.61534 1.00000 −0.864345
1.11 1.67187 1.00000 0.795136 2.68319 1.67187 0 −2.01437 1.00000 4.48594
1.12 1.68292 1.00000 0.832222 −3.40481 1.68292 0 −1.96528 1.00000 −5.73002
1.13 2.13271 1.00000 2.54845 −0.889988 2.13271 0 1.16968 1.00000 −1.89809
1.14 2.69016 1.00000 5.23695 −3.40724 2.69016 0 8.70791 1.00000 −9.16601
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(41\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6027.2.a.bk 14
7.b odd 2 1 6027.2.a.bj 14
7.d odd 6 2 861.2.i.g ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.i.g ✓ 28 7.d odd 6 2
6027.2.a.bj 14 7.b odd 2 1
6027.2.a.bk 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{14} + 2 T_{2}^{13} - 19 T_{2}^{12} - 36 T_{2}^{11} + 134 T_{2}^{10} + 237 T_{2}^{9} - 438 T_{2}^{8} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{14} + 10 T_{5}^{13} + 4 T_{5}^{12} - 253 T_{5}^{11} - 705 T_{5}^{10} + 1475 T_{5}^{9} + \cdots - 166 \) Copy content Toggle raw display
\( T_{13}^{14} + 21 T_{13}^{13} + 89 T_{13}^{12} - 1136 T_{13}^{11} - 12416 T_{13}^{10} - 19825 T_{13}^{9} + \cdots + 50897351 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 10 T^{13} + \cdots - 166 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 16 T^{13} + \cdots + 28802 \) Copy content Toggle raw display
$13$ \( T^{14} + 21 T^{13} + \cdots + 50897351 \) Copy content Toggle raw display
$17$ \( T^{14} + 12 T^{13} + \cdots + 66152 \) Copy content Toggle raw display
$19$ \( T^{14} + 2 T^{13} + \cdots - 320743 \) Copy content Toggle raw display
$23$ \( T^{14} + 7 T^{13} + \cdots - 6955742 \) Copy content Toggle raw display
$29$ \( T^{14} + 16 T^{13} + \cdots + 127408 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 167775313 \) Copy content Toggle raw display
$37$ \( T^{14} - T^{13} + \cdots - 61627267 \) Copy content Toggle raw display
$41$ \( (T + 1)^{14} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 19614000664 \) Copy content Toggle raw display
$47$ \( T^{14} + 12 T^{13} + \cdots + 14289446 \) Copy content Toggle raw display
$53$ \( T^{14} + 20 T^{13} + \cdots - 6984424 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 1078090072 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 205781002708 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 183242489 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 3318503392 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 22867756973 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 5392883753737 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 4861913822 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 3569844798362 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 11760514456 \) Copy content Toggle raw display
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